Collaborative filtering

A distinction is often made between two forms of data collection for recommendation systems. Explicit feedback relies on the user giving explicit signals about their preferences i.e. review ratings. Where as, implicit feedback refers to non-explicit signals of preference e.g. user watch-time. Traditionally, recommender systems can be split into three types:

  • Collaborative filtering (CF): CF produces recommendations based on the knowledge of users’ attitudes towards items, that is, it uses the “wisdom of the crowd” to recommend items.

  • Content-based (CB): CB recommender systems focus on the attributes of the items to recommend other items similar to what the user likes, based on their previous actions or explicit feedback.

  • Hybrid recommendation systems: Hybrid methods are a combination of CB recommending and CF methods

In many applications, content-based features are not easy to extract, and thus, collaborative filtering approaches are preferred. Thus, we will only explore collaborative filtering methods from now on.

CF methods typically fall into three types, memory-based, model-based and more recently deep-learning based (Su & Khoshgoftaar, 2009, He et al., 2017). Neighbour-based CF and item-based/user-based top-N recommendations are typical examples of memory-based systems that utilises user rating data to compute the similarity between users or items. As mentioned previously, common model-based approaches include Bayesian networks, latent semantic models and markov decision processes. In this investigation, we will utilise a weighted matrix factorization approach. Later on, we will generalize the matrix factorization algorithm via a non-linear neural architecture (a softmax model).

However, there are a number of limitations to our approaches such as the inability to model the order of interactions. For instance, Markov chain algorithms (Rendle et al., 2010) can not only encode the same information as traditional CF methods but also the order in which user’s interacted with the items. Furthermore, the sparsity of the frequency matrix (described later on), makes computations prohibitly expensive in real-world settings, without some optimization.

Setup

The next few code cells details the initial preparatory steps needed for the development of our collaborative filtering models, namely importing the required libraries; scaling the ids of users and artists;constructing a indicator variable for presence of user-artist interaction;finding the most assigned tag of an artist.

from __future__ import print_function
import numpy as np
import pandas as pd
import collections
from IPython import display
from matplotlib import pyplot as plt
import sklearn
import sklearn.manifold
import tensorflow.compat.v1 as tf
tf.disable_v2_behavior()
tf.logging.set_verbosity(tf.logging.ERROR)

# Add some convenience functions to Pandas DataFrame.
pd.options.display.max_rows = 10
pd.options.display.float_format = '{:.3f}'.format

# Install Altair and activate its colab renderer.
print("Installing Altair...")
!pip install git+git://github.com/altair-viz/altair.git
import altair as alt
alt.data_transformers.enable('default', max_rows=None)
alt.renderers.enable('colab')
print("Done installing Altair.")
2021-11-28 22:47:25.531632: W tensorflow/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libcudart.so.11.0'; dlerror: libcudart.so.11.0: cannot open shared object file: No such file or directory
2021-11-28 22:47:25.531669: I tensorflow/stream_executor/cuda/cudart_stub.cc:29] Ignore above cudart dlerror if you do not have a GPU set up on your machine.
WARNING:tensorflow:From /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages/tensorflow/python/compat/v2_compat.py:111: disable_resource_variables (from tensorflow.python.ops.variable_scope) is deprecated and will be removed in a future version.
Instructions for updating:
non-resource variables are not supported in the long term
Installing Altair...
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Done installing Altair.
# NEEDED FOR GOOGLE COLAB
# from google.colab import auth
#from google.colab import drive
# import gspread
# from oauth2client.client import GoogleCredentials

# drive.mount('/content/drive/')
# os.chdir("/content/drive/My Drive/DCU/fouth_year/advanced_machine_learning/music-recommodation-system")

Helper functions

def calculate_sparsity(M):
    """
    Computes sparsity of frequency matrix
    """
    matrix_size = len((M['userID'].unique())) * len((M['artistID'].unique()))  # Number of possible interactions in the matrix
    num_plays = len(M['weight']) # Number of weights
    sparsity = (float(num_plays/matrix_size))
    return sparsity
def build_music_sparse_tensor(music_df):
  """
  Args:
    ratings_df: a pd.DataFrame with `userID`, `artistID` and `weight` columns.
    num_rows: an integer representing the number of rows in the frequency matrix
    num_rows: an integer representing the number of columns in the frequency matrix
  Returns:
    a tf.SparseTensor representing the feedback matrix.
  """
  indices = music_df[['userID', 'artistID']].values
  values = music_df['weight'].values
  return tf.SparseTensor(
      indices=indices,
      values=values,
      dense_shape=[num_users, num_artist])
def preproces_ids(music_df):
  """
  Args:
    ratings_df: a pd.DataFrame with `userID`, `artistID` and `weight` columns.
  Returns:
    a pd.DataFrame where userIDs and artistIDs now start at 1 
      and end at n and m (defined above), respectively
    two dictionary preserving the orginal ids. 
  """
  unique_user_ids_list = sorted(music_df['userID'].unique())
  print(unique_user_ids_list[0])

  unique_user_ids = dict(zip(range(0, len(unique_user_ids_list) ),unique_user_ids_list))
  unique_user_ids_switched = dict(zip(unique_user_ids_list, range(0, len(unique_user_ids) )))
  
  unique_artist_ids_list = sorted(music_df['artistID'].unique())
  unique_artist_ids = dict(zip(range(0, len(unique_artist_ids_list) ),unique_artist_ids_list))
  unique_artist_ids_switched = dict(zip(unique_artist_ids_list, range(0, len(unique_artist_ids_list) )))

  music_df['userID'] = music_df['userID'].map(unique_user_ids_switched)
  music_df['artistID'] = music_df['artistID'].map(unique_artist_ids_switched)

  return music_df, unique_user_ids, unique_artist_ids
def split_dataframe(df, holdout_fraction=0.1):
  """Splits a DataFrame into training and test sets.
  Args:
    df: a dataframe.
    holdout_fraction: fraction of dataframe rows to use in the test set.
  Returns:
    train: dataframe for training
    test: dataframe for testing
  """
  test = df.sample(frac=holdout_fraction, replace=False)
  train = df[~df.index.isin(test.index)]
  return train, test

Traditional recommender system development relies on explicit feedback. Many models were designed to tackle this issue as a regression problem. For instance, the input of the model would be a matrix \(F_{nm}\) denoting user’s (m) preference of items (n) on a scale. In the classic movie ratings example, this preference would be users giving a 1-to-5 star rating to different movies.

This dataset contains implicit feedback: that is, observed logs of user interactions with items, in this instance user’s listening counts to artists. However, implicit feedback does not signal negativity, in the same way as a 1-star rating would. In our data, a user could listen to song of an artist a limited number of times. But that does not necessarily mean that the particular user has an aversion to that artist i.e. it could be part of a curated playlist by another user. Therefore, we decide to construct a binary matrix, which has a value of one if the observation is observed (i.e. a listening count has been logged between an artist and a user). Note, a 0 is not used to describe unobserved artist-user interactions. This is for optimization reasons, explained below.

user_artists = pd.read_csv('data/user_artists.dat', sep='\t')
user_artists['weight'] = 1
artists = pd.read_csv('data/artists.dat', sep='\t')
artists.rename({'id':'artistID'}, inplace=True, axis=1)
user_taggedartists = pd.read_csv(r'data/user_taggedartists-timestamps.dat', sep='\t')
user_taggedartists_years = pd.read_csv(r'data/user_taggedartists.dat', sep='\t')
tags = pd.read_csv(open('data/tags.dat', errors='replace'), sep='\t')
user_taggedartists = pd.merge(user_taggedartists, tags, on=['tagID'])
num_users = user_artists.userID.nunique()
num_artist = artists.artistID.nunique()
collab_filter_df = user_artists

Here, we calculate the top 10 tags by popularity. Then, we assign it to a artist, if the artist has a top 10 tag. If an artist’s tags are not in the top 10, we input ‘N/A’. Note, the next cell can take several mintues to compute.

top_10_tags = user_taggedartists['tagValue'].value_counts().index[0:10]
user_taggedartists['top10TagValue'] = None
for index, row in user_taggedartists.iterrows():
  if row['tagValue'] in top_10_tags:
    user_taggedartists.iloc[index, -1] = row['tagValue']
user_taggedartists.fillna('N/A',inplace=True)
artists = pd.merge(user_taggedartists, artists, on=['artistID'], how='right')[['artistID','name','top10TagValue','tagValue']].fillna('N/A')
artists.groupby(['artistID','name','top10TagValue']).agg(lambda x:x.value_counts().index[0]).reset_index()
artists = artists.drop_duplicates(subset=['artistID'])
assert artists.artistID.nunique() == num_artist
artists.rename({'tagValue':'mostCommonGenre'},axis=1, inplace=True)

We require two matrices or embeddings to compute a similarity measure (one for quires and one for items), but how do we get these two embeddings?

Matrix Factorisation

Figure 2: Data flow chart

First, we need to contsruct the feedback matrix \(F \in R^{m \times n}\), where \(m\) is the number of users and \(n\) is the number of artists. The goal is to two generate two lower-dimensional matrices \(U_{mp}\) and \(V_{np}\) ( with \(p << m\) and \(p << n\)), representing latent user and artist components, so that: $\( F \approx UV^\top \)$

First,we attempt to build the frequency matrix for both training and testing data. tf.SparseTensor is used for efficient representation. Three separate arguments are used to represent a tensor, namely indices, values, dense_shape, where a value \(A_{ij} = a\) is encoded by setting indices[k] = [i, j] and values[k] = a. The last tensor dense_shape is used to specify the shape of the full underlying matrix. Note, as the indices arguments represent row and columns indices, some pre-processing needs to be performed on artist and user IDs. The IDs should start from 0 and end at \(m-1\) and \(n-1\) for users and artists respectively. Presently, userIDs start at 2. Two dictionaries, orginal_artist_ids, orginal_user_ids will preserve the original ids for analysis purposes later on. Assertions and print statements are used to ensure the validity of the transformations.

colab_filter_df, orginal_user_ids, orginal_artist_ids =  preproces_ids(collab_filter_df)
2
colab_filter_df.describe()
userID artistID weight
count 92834.000 92834.000 92834.000
mean 944.222 3235.737 1.000
std 546.751 4197.217 0.000
min 0.000 0.000 1.000
25% 470.000 430.000 1.000
50% 944.000 1237.000 1.000
75% 1416.000 4266.000 1.000
max 1891.000 17631.000 1.000

Next, we caulcate the number of unique artists, userids and sparisty of our proposed frequency matrix, before splitting into training and test subsets. Quite a sparse matrix indeed!

print(f'Number of unqiue users are: {collab_filter_df["userID"].nunique()}')
print(f'Number of unqiue artists are: {collab_filter_df["artistID"].nunique()}')
print(f'Sparsity of our frequency matrix: {calculate_sparsity(collab_filter_df)}')
Number of unqiue users are: 1892
Number of unqiue artists are: 17632
Sparsity of our frequency matrix: 0.002782815119924182
collab_filter_df.to_csv('data/test_user_artists.csv',index=False)
frequency_m_train, frequency_m_test = split_dataframe(colab_filter_df)
frequency_m_train_tensor  = build_music_sparse_tensor(frequency_m_train)
frequency_m_test_tensor  = build_music_sparse_tensor(frequency_m_test)
assert num_users  == frequency_m_train_tensor.shape.as_list()[0] 
assert num_artist == frequency_m_train_tensor.shape.as_list()[1] 
assert num_users == frequency_m_test_tensor.shape.as_list()[0] 
assert num_artist == frequency_m_test_tensor.shape.as_list()[1] 

Training a Matrix factorization model

Per the definition above, \(UV^\top\) approximates \(F\). The Mean Squared Error is used to measure this approximation error. In the notation below, k is used to represent the set of observed listening counts, and K is the number of observed listening counts.

\[ \begin{align*} \text{MSE}(F, UV^\top) = \frac{1}{K}\sum_{(i, j) \in k}{( F_{ij} - (UV^\top)_{ij})^2} \end{align*} \]

However, rather than computing the full prediction matrix, \(UV^\top\) and gathering the entries in the embeddings (corresponding to the observed listening counts) , we only gather the embeddings of the observers pairs and compute their dot products. Thereby, we reduce the complexity from \(O(NM)\) to \(O(Kp)\) where \(p\) is the embedding dimension. Stochastic gradient descent (SGD) is used to minimize the loss (objective) function. The SDG algorithim cycles through the observed listening binary and caulates the prediction according to the following equation.

\[ e_{ui} = F_{ui} - U_{i}V_{j} \]

Then it updates the user and artist as embeddings as shown in the following equations.

\[ U_{i} \leftarrow U_{i} + \alpha (e_{ui}V_{j} - \beta U_{i}) \]
\[ V_{j} \leftarrow V_{j} + \alpha (e_{ui}U_{j} - \beta V_{j}) \]

where \(\alpha\) denotes the learning rate. The algorithim continues untill convergence is found.

Other matrix factorization algorithms functions are also commonly used such as Alternating Least Squares (Takács and Tikk, 2012). A modified version of the aforementioned function known as Weighted Alternating Least Squares (WALS) is slower than SDG but can be parallelised. For the purposes of this investigation, we are not particularly concerned with training times/latency requirements so we proceed with SDG.

We also decide to add regularization to our model, to avoid overfitting. Overfitting occurs when the model tries to fit the training dataset to hard and does not generalize well to unseen or future data. In the context of artist recommendation, fitting the observed listening counts often emphasizes learning high similarity (between artists with many listeners), but a good embedding representation also requires learning low similarity (between artists with few listeners).

First, we define the two classes train_matrix_norm and build_matrix_norm class. The build_matrix_norm class computes the necessary pre-processing steps before we train the model such as specifying the loss metric to optimise and the loss components( e.g. gravity loss for the regularized model) and the initial artist and user embeddings. train_matrix_norm simply trains the models and outputs figures detailing the the loss metrics and components. The methods build_vanilla() and build_reg_model() computes the necessary pre-processing steps for the non-regularized and regularized model.

### Training a Matrix Factorization model
class train_matrix_norm(object):
  """Simple class that represents a matrix normalisation model"""
  def __init__(self, embedding_vars, loss, metrics=None):
    """Initializes a Matrix normalisation model 
    Args:
      embedding_vars: A dictionary of tf.Variables.
      loss: A float Tensor. The loss to optimize.
      metrics: optional list of dictionaries of Tensors. The metrics in each
        dictionary will be plotted in a separate figure during training.
    """
    self._embedding_vars = embedding_vars
    self._loss = loss
    self._metrics = metrics
    self._embeddings = {k: None for k in embedding_vars}
    self._session = None


  @property
  def embeddings(self):
    """The embeddings dictionary."""
    return self._embeddings


  def train(self, num_iterations=100, learning_rate=1.0, plot_results=True,
            optimizer=tf.train.GradientDescentOptimizer):
    """Trains the model.
    Args:
      iterations: number of iterations to run.
      learning_rate: optimizer learning rate.
      plot_results: whether to plot the results at the end of training.
      optimizer: the optimizer to use. Default to SDG
    Returns:
      The metrics dictionary evaluated at the last iteration.
    """
    with self._loss.graph.as_default():
      opt = optimizer(learning_rate)
      train_op = opt.minimize(self._loss)
      local_init_op = tf.group(
          tf.variables_initializer(opt.variables()),
          tf.local_variables_initializer())
      if self._session is None:
        self._session = tf.Session()
        with self._session.as_default():
          self._session.run(tf.global_variables_initializer())
          self._session.run(tf.tables_initializer())
          tf.train.start_queue_runners()

    with self._session.as_default():
      local_init_op.run()
      iterations = []
      metrics = self._metrics or ({},)
      metrics_vals = [collections.defaultdict(list) for _ in self._metrics]

      # Train and append results.
      for i in range(num_iterations + 1):
        _, results = self._session.run((train_op, metrics))
        if (i % 10 == 0) or i == num_iterations:
          print("\r iteration %d: " % i + ", ".join(
                ["%s=%f" % (k, v) for r in results for k, v in r.items()]),
                end='')
          iterations.append(i)
          for metric_val, result in zip(metrics_vals, results):
            for k, v in result.items():
              metric_val[k].append(v)

      for k, v in self._embedding_vars.items():
        self._embeddings[k] = v.eval()

      if plot_results:
        # Plot the metrics.
        num_subplots = len(metrics)+1
        fig = plt.figure()
        fig.set_size_inches(num_subplots*10, 8)
        for i, metric_vals in enumerate(metrics_vals):
          ax = fig.add_subplot(1, num_subplots, i+1)
          for k, v in metric_vals.items():
            ax.plot(iterations, v, label=k)
          ax.set_xlim([1, num_iterations])
          ax.legend()
      return results

class build_matrix_norm():
  """Simple class that represents a matrix normalisation model"""
  def __init__(self, listens, embedding_dim=3, regularization_coeff=.1, gravity_coeff=1.,
    init_stddev=0.1):
    """Initializes a Matrix normalisation model 
    Args:
      listens: the DataFrame of artist listening counts.
      embedding_dim: The dimension of the embedding space.
      regularization_coeff: The regularization coefficient lambda.
      gravity_coeff: The gravity regularization coefficient lambda_g.
    Returns:
      A train_matrix_norm object that uses a regularized loss.
  """
    self._embedding_vars = embedding_vars
    self._loss = loss
    self._metrics = metrics
    self._embeddings = {k: None for k in embedding_vars}
    self._session = None

  def sparse_mean_square_error(sparse_listens, user_embeddings, artist_embeddings):
    """
    Args:
      sparse_listens: A SparseTensor rating matrix, of dense_shape [N, M]
      user_embeddings: A dense Tensor U of shape [N, k] where k is the embedding
        dimension, such that U_i is the embedding of user i.
      artist_embeddings: A dense Tensor V of shape [M, k] where k is the embedding
        dimension, such that V_j is the embedding of movie j.
    Returns:
      A scalar Tensor representing the MSE between the true ratings and the
        model's predictions.
    """
    predictions = tf.gather_nd(
        tf.matmul(user_embeddings, artist_embeddings, transpose_b=True),
        sparse_listens.indices)
    loss = tf.losses.mean_squared_error(sparse_listens.values, predictions)
    return loss
    
  def gravity(U, V):
    """Creates a gravity loss given two embedding matrices."""
    return 1. / (U.shape[0].value*V.shape[0].value) * tf.reduce_sum(
        tf.matmul(U, U, transpose_a=True) * tf.matmul(V, V, transpose_a=True))
  
  def build_vanilla(embedding_dim=3, init_stddev=1.):
    """performs the necessary preprocessing steps for the regularized model.  """ 
    # Initialize the embeddings using a normal distribution.
    U = tf.Variable(tf.random.normal(
      [frequency_m_train_tensor.dense_shape[0], embedding_dim], stddev=init_stddev))
    V = tf.Variable(tf.random.normal(
      [frequency_m_train_tensor.dense_shape[1], embedding_dim], stddev=init_stddev))
    
    embeddings = {"userID": U, "artistID": V}
    error_train = build_matrix_norm.sparse_mean_square_error(frequency_m_train_tensor, U, V)
    error_test = build_matrix_norm.sparse_mean_square_error(frequency_m_test_tensor, U, V)
    metrics = {
        'train_error': error_train,
        'test_error': error_test
    }
    return train_matrix_norm(embeddings, error_train, [metrics])


  def build_reg_model(embedding_dim=3, regularization_coeff=.1, gravity_coeff=1.,
  init_stddev=0.1
  ):
    """performs the necessary preprocessing steps for the regularized model.  """ 
    U = tf.Variable(tf.random.normal(
      [frequency_m_train_tensor.dense_shape[0], embedding_dim], stddev=init_stddev))
    V = tf.Variable(tf.random.normal(
      [frequency_m_train_tensor.dense_shape[1], embedding_dim], stddev=init_stddev))
  
    embeddings = {"userID": U, "artistID": V}

    error_train = build_matrix_norm.sparse_mean_square_error(frequency_m_train_tensor, U, V)
    error_test = build_matrix_norm.sparse_mean_square_error(frequency_m_test_tensor, U, V)
    gravity_loss = gravity_coeff * build_matrix_norm.gravity(U, V)
    regularization_loss = regularization_coeff * (
      tf.reduce_sum(U*U)/U.shape[0].value + tf.reduce_sum(V*V)/V.shape[0].value)
    total_loss = error_train + regularization_loss + gravity_loss
    losses = {
      'train_error_observed': error_train,
      'test_error_observed': error_test,
    }
    loss_components = {
      'observed_loss': error_train,
      'regularization_loss': regularization_loss,
      'gravity_loss': gravity_loss,
    }
    #embeddings = {"userID": U, "artistID": V}

    return train_matrix_norm(embeddings, total_loss, [losses, loss_components])

Vanilla Model (non-regularized)

vanilla_model = build_matrix_norm.build_vanilla(embedding_dim=35,init_stddev=.05)
vanilla_model.train(num_iterations=2000, learning_rate=20.)
2021-11-28 22:49:54.706908: W tensorflow/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libcuda.so.1'; dlerror: libcuda.so.1: cannot open shared object file: No such file or directory
2021-11-28 22:49:54.706954: W tensorflow/stream_executor/cuda/cuda_driver.cc:269] failed call to cuInit: UNKNOWN ERROR (303)
2021-11-28 22:49:54.706980: I tensorflow/stream_executor/cuda/cuda_diagnostics.cc:156] kernel driver does not appear to be running on this host (fv-az42-909): /proc/driver/nvidia/version does not exist
2021-11-28 22:49:54.707279: I tensorflow/core/platform/cpu_feature_guard.cc:151] This TensorFlow binary is optimized with oneAPI Deep Neural Network Library (oneDNN) to use the following CPU instructions in performance-critical operations:  AVX2 AVX512F FMA
To enable them in other operations, rebuild TensorFlow with the appropriate compiler flags.
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[{'train_error': 0.0013244184, 'test_error': 0.14924659}]
_images/Collaborative_filtering_27_203.png

Regularized moodel

reg_model = build_matrix_norm.build_reg_model(regularization_coeff=0.1, gravity_coeff=1.0, embedding_dim=35,init_stddev=.05)
reg_model.train(num_iterations=2000, learning_rate=20.)
 iteration 0: train_error_observed=1.000341, test_error_observed=1.000450, observed_loss=1.000341, regularization_loss=0.017547, gravity_loss=0.000220
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 iteration 1910: train_error_observed=0.094414, test_error_observed=0.241462, observed_loss=0.094414, regularization_loss=0.114956, gravity_loss=0.048365
 iteration 1920: train_error_observed=0.094036, test_error_observed=0.241406, observed_loss=0.094036, regularization_loss=0.115112, gravity_loss=0.048294
 iteration 1930: train_error_observed=0.093663, test_error_observed=0.241352, observed_loss=0.093663, regularization_loss=0.115267, gravity_loss=0.048223
 iteration 1940: train_error_observed=0.093293, test_error_observed=0.241299, observed_loss=0.093293, regularization_loss=0.115420, gravity_loss=0.048153
 iteration 1950: train_error_observed=0.092927, test_error_observed=0.241247, observed_loss=0.092927, regularization_loss=0.115572, gravity_loss=0.048084
 iteration 1960: train_error_observed=0.092565, test_error_observed=0.241196, observed_loss=0.092565, regularization_loss=0.115723, gravity_loss=0.048014
 iteration 1970: train_error_observed=0.092207, test_error_observed=0.241146, observed_loss=0.092207, regularization_loss=0.115873, gravity_loss=0.047945
 iteration 1980: train_error_observed=0.091853, test_error_observed=0.241097, observed_loss=0.091853, regularization_loss=0.116021, gravity_loss=0.047877
 iteration 1990: train_error_observed=0.091502, test_error_observed=0.241049, observed_loss=0.091502, regularization_loss=0.116168, gravity_loss=0.047808
 iteration 2000: train_error_observed=0.091155, test_error_observed=0.241002, observed_loss=0.091155, regularization_loss=0.116314, gravity_loss=0.047740
[{'train_error_observed': 0.09115525, 'test_error_observed': 0.24100238},
 {'observed_loss': 0.09115525,
  'regularization_loss': 0.11631441,
  'gravity_loss': 0.047740445}]
_images/Collaborative_filtering_30_202.png

In both models, we observe a steep loss in train error and test as the model progress. Although, the regularized model has a higher MSE, both on the training and test set. It must be noted that the quality of recommendation is improved when regularization is added, which is proven when the artist_neighbors() function is utilized. In addition, we observe in the end evaluation section, that the the performance of the model is improved when regularization is added. The test error decreases similarity to the test error, although it plateaus around the 1000 epoch mark. As expected, the the additional loss generated by the regularization functions increases over epochs. We add the following regularisation terms to our model.

  • Regularization of the model parameters. This is a common \(\ell_2\) regularization term on the embedding matrices, given by \(r(U, V) = \frac{1}{N} \sum_i \|U_i\|^2 + \frac{1}{M}\sum_j \|V_j\|^2\).

  • A global prior that pushes the prediction of any pair towards zero, called the gravity term. This is given by \(g(U, V) = \frac{1}{MN} \sum_{i = 1}^N \sum_{j = 1}^M \langle U_i, V_j \rangle^2\)

These terms modifies the “global” loss (as in, the sum of the network loss and the regularization loss) in order to drive the optimization algorithm in desired directions i.e. prevent overfitting.

Evaluating the embeddings

We will use two similairty meausres to inspect the robustness of our system:

  • Dot product: score of artist j \(\langle u, V_j \rangle\).

  • Cosine angle: score of artist j \(\frac{\langle u, V_j \rangle}{\|u\|\|V_j\|}\).

DOT = 'dot'
COSINE = 'cosine'
def compute_scores(query_embedding, item_embeddings, measure=DOT):
  """Computes the scores of the candidates given a query.
  Args:
    query_embedding: a vector of shape [k], representing the query embedding.
    item_embeddings: a matrix of shape [N, k], such that row i is the embedding
      of item i.
    measure: a string specifying the similarity measure to be used. Can be
      either DOT or COSINE.
  Returns:
    scores: a vector of shape [N], such that scores[i] is the score of item i.
  """
  u = query_embedding
  V = item_embeddings
  if measure == COSINE:
    V = V / np.linalg.norm(V, axis=1, keepdims=True)
    u = u / np.linalg.norm(u)
  scores = u.dot(V.T)
  return scores
def user_recommendations(model,user_id, k=15, measure=DOT, exclude_rated=False):
    scores = compute_scores(
        model.embeddings["userID"][user_id], model.embeddings["artistID"], measure)
    score_key = measure + ' score'
    df = pd.DataFrame({
        'score': list(scores),
        'name': artists.sort_values('artistID', ascending=True)['name'],
        'most assigned tag':artists.sort_values('artistID', ascending=True)['mostCommonGenre']
    })
    return df.sort_values(['score'], ascending=False).head(k)


def artist_neighbors(model, title_substring, measure=DOT, k=6):
  # Search for artist ids that match the given substring.
  inv_artist_id_mapping = {v: k for k, v in orginal_artist_ids.items()}
  ids =  artists[artists['name'].str.contains(title_substring)].artistID.values
  titles = artists[artists.artistID.isin(ids)]['name'].values
  if len(titles) == 0:
    raise ValueError("Found no artists with name %s" % title_substring)
  print("Nearest neighbors of : %s." % titles[0])
  if len(titles) > 1:
    print("[Found more than one matching artist. Other candidates: {}]".format(
        ", ".join(titles[1:])))
  artists_id_orginal = ids[0]
  asrtists_id_mapped = inv_artist_id_mapping[ids[0]]
  scores = compute_scores(
      model.embeddings["artistID"][asrtists_id_mapped], model.embeddings["artistID"],
      measure)
  score_key = measure + ' score'
  df = pd.DataFrame({
      score_key: list(scores),
      'name': artists.sort_values('artistID', ascending=True)['name'],
      'most assigned tag':artists.sort_values('artistID', ascending=True)['mostCommonGenre']
  })
  return df.sort_values([score_key], ascending=False).head(k)

Here, we find the most similar artists to the band the cure. We also include the most assigned tag associated with an artist. The reccomdations are conistent with our domain knowedge of bands similar to the cure.

artist_neighbors(vanilla_model, "The Cure", DOT)
Nearest neighbors of : The Cure.
dot score name most assigned tag
9437 0.546 The Cure chillout
84188 0.531 Pato Fu pop
57843 0.531 Nightwish pop
71605 0.530 Guns N' Roses atmospheric
17278 0.530 Kings of Leon chillout
58990 0.530 Queen 80's
artist_neighbors(vanilla_model, "The Cure", COSINE)
Nearest neighbors of : The Cure.
cosine score name most assigned tag
9437 1.000 The Cure chillout
71605 0.959 Guns N' Roses atmospheric
8273 0.957 Radiohead chillout
58990 0.953 Queen 80's
15847 0.951 Red Hot Chili Peppers chillout
10850 0.951 Placebo chillout
artist_neighbors(reg_model, "The Cure", DOT)
Nearest neighbors of : The Cure.
dot score name most assigned tag
16680 3.261 The Beatles chillout
9437 3.185 The Cure chillout
12363 3.176 Muse chillout
3259 3.166 Coldplay chillout
18364 3.163 Nirvana pop
15847 3.147 Red Hot Chili Peppers chillout
artist_neighbors(reg_model, "The Cure", COSINE)
Nearest neighbors of : The Cure.
cosine score name most assigned tag
9437 1.000 The Cure chillout
43413 0.963 David Bowie chillout
8273 0.958 Radiohead chillout
32942 0.958 The Smiths groove
38968 0.958 U2 electronic
4936 0.952 Depeche Mode chillout

We observe that dot product tends to recommends more popular artists such as Nirvana and The Beatles, where as Cosine Similarity recommends more obscure artists. This is likely due to the fact that the norm of the embedding in matrix factorization is often correlated with prevalence. The regularised model seems to output better reccomodations as the varation of the most assigned tag attribute is less when compared to the vanilla model. In addition, Marilyn Manson was recommended by the vanilla model in our intial run. We argue that these artists are most dis-similar! However, this observation is subject to change when you run the model, as we initialize the embedddings with a random gaussian generator.

def artist_embedding_norm(models):
  """Visualizes the norm and number of ratings of the artist embeddings.
  Args:
    model: A train_matrix_norm object.
  """
  if not isinstance(models, list):
    models = [models]
    df = pd.DataFrame({
          'name': artists.sort_values('artistID', ascending=True)['name'].values,
        'number of user-artist interactions': user_artists[['artistID','userID']].sort_values('artistID', ascending=True).groupby('artistID').count()['userID'].values,
    })
  charts = []
  brush = alt.selection_interval()
  for i, model in enumerate(models):
    norm_key = 'norm'+str(i)
    df[norm_key] = np.linalg.norm(model.embeddings["artistID"], axis=1)
    nearest = alt.selection(
        type='single', encodings=['x', 'y'], on='mouseover', nearest=True,
        empty='none')
    base = alt.Chart().mark_circle().encode(
        x='number of user-artist interactions',
        y=norm_key,
        color=alt.condition(brush, alt.value('#4c78a8'), alt.value('lightgray'))
    ).properties(
        selection=nearest).add_selection(brush)
    text = alt.Chart().mark_text(align='center', dx=5, dy=-5).encode(
        x='number of user-artist interactions', y=norm_key,
        text=alt.condition(nearest, 'name', alt.value('')))
    charts.append(alt.layer(base, text))
  return alt.hconcat(*charts, data=df)

artist_embedding_norm(reg_model)
def visualize_movie_embeddings(data, x, y):
  genre_filter = alt.selection_multi(fields=['top10TagValue'])
  genre_chart = alt.Chart().mark_bar().encode(
      x="count()",
      y=alt.Y('top10TagValue'),
      color=alt.condition(
          genre_filter,
          alt.Color("top10TagValue:N"),
          alt.value('lightgray'))
  ).properties(height=300, selection=genre_filter)
  nearest = alt.selection(
      type='single', encodings=['x', 'y'], on='mouseover', nearest=True,
      empty='none')
  base = alt.Chart().mark_circle().encode(
      x=x,
      y=y,
      color=alt.condition(genre_filter, "top10TagValue", alt.value("whitesmoke")),
  ).properties(
      width=600,
      height=600,
      selection=nearest)
  text = alt.Chart().mark_text(align='left', dx=5, dy=-5).encode(
      x=x,
      y=y,
      text=alt.condition(nearest, 'name', alt.value('')))
  return alt.hconcat(alt.layer(base, text), genre_chart, data=data)

def tsne_movie_embeddings(model):
  """Visualizes the movie embeddings, projected using t-SNE with Cosine measure.
  Args:
    model: A MFModel object.
  """
  tsne = sklearn.manifold.TSNE(
      n_components=2, perplexity=40, metric='cosine', early_exaggeration=10.0,
      init='pca', verbose=True, n_iter=400)

  print('Running t-SNE...')
  V_proj = tsne.fit_transform(model.embeddings["artistID"])
  artists.loc[:,'x'] = V_proj[:, 0]
  artists.loc[:,'y'] = V_proj[:, 1]
  return visualize_movie_embeddings(artists, 'x', 'y')

T-distributed stochastic neighbor embedding (t-SNE) is a dimensionality reduction algorithm useful for visualizing high dimensional data. We use this algorithim to visualise our embeddings of the regualrised model. Due to the large number of user submitted semantic categories, we decide to color-code the top 15 tags, with the rest being labelled as ‘N/A’. Although the sea of orange, indicating’N/A’, makes it difficult to interrupt these results, the regularised model seems to adequaltly cluster artists of a similar genre in it’s embeddings.

tsne_movie_embeddings(reg_model)
Running t-SNE...
[t-SNE] Computing 121 nearest neighbors...
[t-SNE] Indexed 17632 samples in 0.001s...
/opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages/sklearn/manifold/_t_sne.py:793: FutureWarning: The default learning rate in TSNE will change from 200.0 to 'auto' in 1.2.
  FutureWarning,
/opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages/sklearn/manifold/_t_sne.py:827: FutureWarning: 'square_distances' has been introduced in 0.24 to help phase out legacy squaring behavior. The 'legacy' setting will be removed in 1.1 (renaming of 0.26), and the default setting will be changed to True. In 1.3, 'square_distances' will be removed altogether, and distances will be squared by default. Set 'square_distances'=True to silence this warning.
  FutureWarning,
[t-SNE] Computed neighbors for 17632 samples in 5.322s...
[t-SNE] Computed conditional probabilities for sample 1000 / 17632
[t-SNE] Computed conditional probabilities for sample 2000 / 17632
[t-SNE] Computed conditional probabilities for sample 3000 / 17632
[t-SNE] Computed conditional probabilities for sample 4000 / 17632
[t-SNE] Computed conditional probabilities for sample 5000 / 17632
[t-SNE] Computed conditional probabilities for sample 6000 / 17632
[t-SNE] Computed conditional probabilities for sample 7000 / 17632
[t-SNE] Computed conditional probabilities for sample 8000 / 17632
[t-SNE] Computed conditional probabilities for sample 9000 / 17632
[t-SNE] Computed conditional probabilities for sample 10000 / 17632
[t-SNE] Computed conditional probabilities for sample 11000 / 17632
[t-SNE] Computed conditional probabilities for sample 12000 / 17632
[t-SNE] Computed conditional probabilities for sample 13000 / 17632
[t-SNE] Computed conditional probabilities for sample 14000 / 17632
[t-SNE] Computed conditional probabilities for sample 15000 / 17632
[t-SNE] Computed conditional probabilities for sample 16000 / 17632
[t-SNE] Computed conditional probabilities for sample 17000 / 17632
[t-SNE] Computed conditional probabilities for sample 17632 / 17632
[t-SNE] Mean sigma: 0.179662
/opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages/sklearn/manifold/_t_sne.py:986: FutureWarning: The PCA initialization in TSNE will change to have the standard deviation of PC1 equal to 1e-4 in 1.2. This will ensure better convergence.
  FutureWarning,
[t-SNE] KL divergence after 250 iterations with early exaggeration: 77.142731
[t-SNE] KL divergence after 400 iterations: 2.762864
def m_embedding_norm(models):
  """Visualizes the norm and number of ratings of the movie embeddings.
  Args:
    model: A MFModel object.
  """
  if not isinstance(models, list):
    models = [models]
    df = pd.DataFrame({
          'title': artists.sort_values('artistID', ascending=True)['name'].values,
        'num_ratings': user_artists[['artistID','userID']].sort_values('artistID', ascending=True).groupby('artistID').count()['userID'].values,
    })
  charts = []
  brush = alt.selection_interval()
  for i, model in enumerate(models):
    norm_key = 'norm'+str(i)
    df[norm_key] = np.linalg.norm(model.embeddings["artistID"], axis=1)
    nearest = alt.selection(
        type='single', encodings=['x', 'y'], on='mouseover', nearest=True,
        empty='none')
    base = alt.Chart().mark_circle().encode(
        x='num_ratings',
        y=norm_key,
        color=alt.condition(brush, alt.value('#4c78a8'), alt.value('lightgray'))
    ).properties(
        selection=nearest).add_selection(brush)
    text = alt.Chart().mark_text(align='center', dx=5, dy=-5).encode(
        x='num_ratings', y=norm_key,
        text=alt.condition(nearest, 'title', alt.value('')))
    charts.append(alt.layer(base, text))
  return alt.hconcat(*charts, data=df)

Demo

You can find the most similar artist to a specified artist (that is contained in Last.FM) using the artist_neighbours() function. Similarily, you can find the top 10 recommendations of a particular userID [0 to 1891] using the user_recommendations() function. The first argument specifies the desired model, second argument the userID and third the top-k recommendations. Fourth argument represents the similarity measure, either DOT or COSINE (default = DOT, not a string).

user_recommendations(reg_model, 234, 10, COSINE)
score name most assigned tag
126400 0.933 The Vibrators punk
126582 0.914 Validuaté N/A
126554 0.912 Moreira da Silva N/A
126491 0.902 Bandas Gaúchas - www.DownsMtv.com N/A
126513 0.897 Graforréia Xilarmônica rock
126539 0.892 Menstruação Anarquika N/A
126451 0.880 Tim Maia pop
126583 0.848 The Saints punk
126490 0.833 Street Bulldogs N/A
126536 0.827 S.O.D. thrash metal

To further demonstrate the robustness of the system and measure the serendipity of our model, we incorporate the top artists that we listen to on Spotify (i.e. an unknown user). Note, these artists have to also be in the Last.FM dataset. The recommendation system should output similar artists based on it’s artist embeddings. The Spotipy library is used to interact with Spotify’s API. The similarity measure used is the Dot product. Due to the short lived nature of the spotify token and the fact you have to sign into a pop-up to retrieve the authentication token, we simply list our top 5 artists manually. If we did not, jupyter book will stall when attempting to build as it is waiting for our response. However, we provide the code used to retrieve the short-lived token for verification purposes.

"""
import spotipy
from spotipy.oauth2 import SpotifyOAuth
client_id = <insert_your_client_id>
client_secret = <insert your client secret>
redirect_url = '<insert your redirect uri>
scope = "user-top-read user-read-playback-state streaming ugc-image-upload playlist-modify-public"

authenticate_manager = spotipy.oauth2.SpotifyOAuth(client_id = client_id,client_secret = client_secret,redirect_uri =redirect_url,scope =scope,show_dialog = True)
sp = spotipy.Spotify(auth_manager=authenticate_manager)

artists_long = sp.current_user_top_artists(limit=5, time_range="long_term")
"""
top_5_artists =[
                 'Coldplay',
                 'Paramore',
                 'Arctic Monkeys',
                 'Lily Allen',
                 'Miley Cyrus'
]
spotify_reccomdations_df = pd.DataFrame()
for artist in top_5_artists:
  similar_artist_df = artist_neighbors(reg_model, artist)[['name','dot score']]
  spotify_reccomdations_df = pd.concat([spotify_reccomdations_df, similar_artist_df])
spotify_reccomdations_df.sort_values('dot score', ascending=False).head(10)
Nearest neighbors of : Coldplay.
[Found more than one matching artist. Other candidates: Jay-Z & Coldplay, Coldplay/U2]
Nearest neighbors of : Paramore.
[Found more than one matching artist. Other candidates: Paramore攀]
Nearest neighbors of : Arctic Monkeys.
[Found more than one matching artist. Other candidates: Arctic Monkeys vs The Killers]
Nearest neighbors of : Lily Allen.
Nearest neighbors of : Miley Cyrus.
[Found more than one matching artist. Other candidates: Miley Cyrus攀, Demi Lovato Ft. Miley Cyrus Ft. Selena Gomez Ft. Jonas Brothers, Miley Cyrus and Billy Ray Cyrus, Miley Cyrus and John Travolta, Hannah Montana and Miley Cyrus]
name dot score
3259 Coldplay 3.670
12363 Muse 3.595
37842 Paramore 3.537
17472 The Killers 3.531
24447 Lily Allen 3.486
6543 Lady Gaga 3.477
6543 Lady Gaga 3.476
6543 Lady Gaga 3.474
17832 Green Day 3.466
16680 The Beatles 3.464

We believe these recommodations are good as when our model was given an artist in the top five, it actually recommended other artits in the top five.

Evaluation Code

This is the code needed to produce the in-depth model comparison. As we decided to use different notebooks for different models, the results of this code will be combined and explained later in the book.

## create holdout test set for each user (15 items)
user_artists = pd.read_csv('data/user_artists.dat', sep='\t')
user_ids = []
holdout_artits = []
for user_id in user_artists.userID.unique():
  top_15_artists = user_artists[user_artists.userID == user_id].sort_values(by='weight').head(15).artistID.tolist()
  if len(top_15_artists) == 15:
    holdout_artits.append(top_15_artists)
    user_ids.append(user_id)
holdout_df = pd.DataFrame(data={'userID':user_ids,'holdout_artists':holdout_artits})

holdout_df.to_csv('data/evaluation/test-set.csv',index=False)
## Finding the models vanilla, regualrised predection for each user. 
def get_top_15_model_predictions(model, measure):
  """Computes the top 15 predictions for a given model
  Args:
    model: the name of the model
    measure: a string specifying the similarity measure to be used. Can be
      either DOT or COSINE.
  Returns:
    predicted_df a dataframe containing userIDs, their top 15 artists by the model, and the correspnding scores.
  """
  artist_name_id_dict = dict(zip(artists['name'], artists['artistID']))
  user_ids = []
  predicted_artists = []
  scores_list = []
  for new_user_id, orginal_user_id in orginal_user_ids.items():
    top_15_names = user_recommendations(model, new_user_id, k=15,measure=measure )['name'].values
    top_15_scores = user_recommendations(model, new_user_id, k=15, measure=measure )['score'].values.tolist()
    artist_ids = []
    for name in top_15_names:
      artist_ids.append(artist_name_id_dict[name])
    predicted_artists.append(artist_ids)
    user_ids.append(orginal_user_id)
    scores_list.append(top_15_scores)
  predicted_df = pd.DataFrame(data={'userID':user_ids,'predictions_artists':predicted_artists, 'score':scores_list })
  return predicted_df
# save the recommended artits into dfs and save them to data/evaluation folder
vanilla_dot_pred= get_top_15_model_predictions(vanilla_model, measure=DOT)
vanilla_cos_pred = get_top_15_model_predictions(vanilla_model, measure=COSINE)
reg_dot_pred= get_top_15_model_predictions(reg_model, measure=DOT)
reg_cos_pred = get_top_15_model_predictions(reg_model, measure=COSINE)

vanilla_dot_pred.to_csv('data/evaluation/vannila_dot_pred.csv',index=False)
vanilla_cos_pred.to_csv('data/evaluation/vanila_cos_pred.csv',index=False)
reg_dot_pred.to_csv('data/evaluation/reg_dot_pred.csv',index=False)
reg_cos_pred.to_csv('data/evaluation/reg_cos_pred.csv',index=False)